Editing Generating function (section)

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''<sup>[∞]</sup>(''z'')}}, expanded by:
<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''P<sub>h</sub>''(''z'')}} and {{math|''Q<sub>h</sub>''(''z'')}}, defined recursively by:
<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv<sub>''h''</sub>(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''j<sub>n</sub>''}}, ''and'' for {{math|''M<sub>h</sub>'' ≔ ab<sub>2</sub> ⋯ ab<sub>''h'' + 1</sub>}} if {{math|''h'' ‖  ''M''<sub>''h''</sub>}} then we have the congruence
<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab<sub>''i''</sub>}<nowiki/>}} and {{math|{''c''<sub>''i''</sub>}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} as in the examples contained in the table below.